The thing that's always bugged me, across all editions:
The difference between being fully functional at 1 h.p. and dead (in 0e-1e*) dying (in 1e*-2e-3e-5e) or down (4e) at 0 h.p.; and that that one last little hit point holds so much more relevance than any of the other 37 you just lost.
There absolutely need to be some mechanics shoehorned in between those two numbers - 1 and 0 - to cover and allow for and codify the huge range of conditions that exist between "fully functional" and "dying" or "dead". Some - including our own crew - have tried, adding body points or a wounded condition or a consciousness check or whatever, to varying degrees of success. 4e waved at it with 'bloodied', and would have been really on to something except far too many times being bloodied gave benefits rather than penalties.
There's some design space here for this stuff if one allows for negative hit points but alas, 5e threw that out. Reviving and greatly augmenting the 4e bloodied mechanic such that you're hurt (with mechanical ramifications) at half h.p., seriously hurt (with further ramifications) at 1/4 h.p. and so forth could work. Now, on seeing this some will doubtless howl "death spiral!" in protest. Too bad. A death spiral - where the worse off you are the worse things get for you - is the whole damn point; and easily solved by simply keeping your hit points topped up.
A pleasant side effect here would be a reduction of the whack-a-mole problem which, while 5e catches a lot of flak for, has existed in some form ever since 1e-2e went to death at -10. Here, while you could still cure someone up from 0 or below and get them standing, their effectiveness would still be reduced and it'd take a lot more curing to get them to full functionality (i.e. above half h.p.).
* - 1e kinda split between death at 0 and death at -10, so it counts as both here.